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76 relations: Absolute value, Baire category theorem, Baire space, Banach fixed-point theorem, Banach space, Bounded function, Cauchy sequence, Cauchy space, Closed set, Compact convergence, Compact space, Completely metrizable space, Completely uniformizable space, Completion (algebra), Complex number, Construction of the real numbers, Continuous function, Continuous functions on a compact Hausdorff space, Countable set, Dense set, Discrete space, Distinct (mathematics), Empty set, Equivalence class, Equivalence relation, Euclidean distance, Euclidean space, Field (mathematics), Fréchet space, Georg Cantor, Heine–Borel theorem, Hilbert space, Homeomorphism, If and only if, Infimum and supremum, Inner product space, Interior (topology), Intersection (set theory), Interval (mathematics), Inverse function theorem, Irrational number, Isometry, Knaster–Tarski theorem, Limit of a sequence, Locally convex topological vector space, Mathematical analysis, Metric space, Net (mathematics), Normed vector space, Nowhere dense set, ..., P-adic number, Pacific Journal of Mathematics, Polish space, Prime number, Product topology, Pseudometric space, Rational number, Real number, Sequence, Set (mathematics), Square root of 2, Subtraction, Topological group, Topological space, Topological vector space, Topology, Total order, Totally bounded space, Uniform continuity, Uniform norm, Uniform space, Union (set theory), Uniqueness quantification, Unit interval, Universal property, Up to. Expand index (26 more) »
Absolute value
In mathematics, the absolute value or modulus of a real number is the non-negative value of without regard to its sign.
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Baire category theorem
The Baire category theorem (BCT) is an important tool in general topology and functional analysis.
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Baire space
In mathematics, a Baire space is a topological space such that every intersection of a countable collection of open dense sets in the space is also dense.
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Banach fixed-point theorem
In mathematics, the Banach fixed-point theorem (also known as the contraction mapping theorem or contraction mapping principle) is an important tool in the theory of metric spaces; it guarantees the existence and uniqueness of fixed points of certain self-maps of metric spaces, and provides a constructive method to find those fixed points.
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Banach space
In mathematics, more specifically in functional analysis, a Banach space (pronounced) is a complete normed vector space.
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Bounded function
In mathematics, a function f defined on some set X with real or complex values is called bounded, if the set of its values is bounded.
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Cauchy sequence
In mathematics, a Cauchy sequence, named after Augustin-Louis Cauchy, is a sequence whose elements become arbitrarily close to each other as the sequence progresses.
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Cauchy space
In general topology and analysis, a Cauchy space is a generalization of metric spaces and uniform spaces for which the notion of Cauchy convergence still makes sense.
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Closed set
In geometry, topology, and related branches of mathematics, a closed set is a set whose complement is an open set.
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Compact convergence
In mathematics compact convergence (or uniform convergence on compact sets) is a type of convergence which generalizes the idea of uniform convergence.
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Compact space
In mathematics, and more specifically in general topology, compactness is a property that generalizes the notion of a subset of Euclidean space being closed (that is, containing all its limit points) and bounded (that is, having all its points lie within some fixed distance of each other).
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Completely metrizable space
In mathematics, a completely metrizable space (metrically topologically complete space) is a topological space (X, T) for which there exists at least one metric d on X such that (X, d) is a complete metric space and d induces the topology T. The term topologically complete space is employed by some authors as a synonym for completely metrizable space, but sometimes also used for other classes of topological spaces, like completely uniformizable spaces or Čech-complete spaces.
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Completely uniformizable space
In mathematics, a topological space (X, T) is called completely uniformizable (or Dieudonné complete) if there exists at least one complete uniformity that induces the topology T. Some authors additionally require X to be Hausdorff.
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Completion (algebra)
In abstract algebra, a completion is any of several related functors on rings and modules that result in complete topological rings and modules.
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Complex number
A complex number is a number that can be expressed in the form, where and are real numbers, and is a solution of the equation.
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Construction of the real numbers
In mathematics, there are several ways of defining the real number system as an ordered field.
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Continuous function
In mathematics, a continuous function is a function for which sufficiently small changes in the input result in arbitrarily small changes in the output.
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Continuous functions on a compact Hausdorff space
In mathematical analysis, and especially functional analysis, a fundamental role is played by the space of continuous functions on a compact Hausdorff space with values in the real or complex numbers.
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Countable set
In mathematics, a countable set is a set with the same cardinality (number of elements) as some subset of the set of natural numbers.
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Dense set
In topology and related areas of mathematics, a subset A of a topological space X is called dense (in X) if every point x in X either belongs to A or is a limit point of A, that is the closure of A is constituting the whole set X. Informally, for every point in X, the point is either in A or arbitrarily "close" to a member of A — for instance, every real number either is a rational number or has a rational number arbitrarily close to it (see Diophantine approximation).
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Discrete space
In topology, a discrete space is a particularly simple example of a topological space or similar structure, one in which the points form a discontinuous sequence, meaning they are isolated from each other in a certain sense.
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Distinct (mathematics)
In mathematics, two things are called distinct if they are not equal.
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Empty set
In mathematics, and more specifically set theory, the empty set or null set is the unique set having no elements; its size or cardinality (count of elements in a set) is zero.
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Equivalence class
In mathematics, when the elements of some set S have a notion of equivalence (formalized as an equivalence relation) defined on them, then one may naturally split the set S into equivalence classes.
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Equivalence relation
In mathematics, an equivalence relation is a binary relation that is reflexive, symmetric and transitive.
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Euclidean distance
In mathematics, the Euclidean distance or Euclidean metric is the "ordinary" straight-line distance between two points in Euclidean space.
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Euclidean space
In geometry, Euclidean space encompasses the two-dimensional Euclidean plane, the three-dimensional space of Euclidean geometry, and certain other spaces.
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Field (mathematics)
In mathematics, a field is a set on which addition, subtraction, multiplication, and division are defined, and behave as when they are applied to rational and real numbers.
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Fréchet space
In functional analysis and related areas of mathematics, Fréchet spaces, named after Maurice Fréchet, are special topological vector spaces.
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Georg Cantor
Georg Ferdinand Ludwig Philipp Cantor (– January 6, 1918) was a German mathematician.
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Heine–Borel theorem
In real analysis the Heine–Borel theorem, named after Eduard Heine and Émile Borel, states: For a subset S of Euclidean space Rn, the following two statements are equivalent.
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Hilbert space
The mathematical concept of a Hilbert space, named after David Hilbert, generalizes the notion of Euclidean space.
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Homeomorphism
In the mathematical field of topology, a homeomorphism or topological isomorphism or bi continuous function is a continuous function between topological spaces that has a continuous inverse function.
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If and only if
In logic and related fields such as mathematics and philosophy, if and only if (shortened iff) is a biconditional logical connective between statements.
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Infimum and supremum
In mathematics, the infimum (abbreviated inf; plural infima) of a subset S of a partially ordered set T is the greatest element in T that is less than or equal to all elements of S, if such an element exists.
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Inner product space
In linear algebra, an inner product space is a vector space with an additional structure called an inner product.
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Interior (topology)
In mathematics, specifically in topology, the interior of a subset S of points of a topological space X consists of all points of S that do not belong to the boundary of S. A point that is in the interior of S is an interior point of S. The interior of S is the complement of the closure of the complement of S. In this sense interior and closure are dual notions.
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Intersection (set theory)
In mathematics, the intersection A ∩ B of two sets A and B is the set that contains all elements of A that also belong to B (or equivalently, all elements of B that also belong to A), but no other elements.
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Interval (mathematics)
In mathematics, a (real) interval is a set of real numbers with the property that any number that lies between two numbers in the set is also included in the set.
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Inverse function theorem
In mathematics, specifically differential calculus, the inverse function theorem gives a sufficient condition for a function to be invertible in a neighborhood of a point in its domain: namely, that its derivative is continuous and non-zero at the point.
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Irrational number
In mathematics, the irrational numbers are all the real numbers which are not rational numbers, the latter being the numbers constructed from ratios (or fractions) of integers.
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Isometry
In mathematics, an isometry (or congruence, or congruent transformation) is a distance-preserving transformation between metric spaces, usually assumed to be bijective.
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Knaster–Tarski theorem
In the mathematical areas of order and lattice theory, the Knaster–Tarski theorem, named after Bronisław Knaster and Alfred Tarski, states the following: It was Tarski who stated the result in its most general form, and so the theorem is often known as Tarski's fixed point theorem.
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Limit of a sequence
As the positive integer n becomes larger and larger, the value n\cdot \sin\bigg(\frac1\bigg) becomes arbitrarily close to 1.
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Locally convex topological vector space
In functional analysis and related areas of mathematics, locally convex topological vector spaces or locally convex spaces are examples of topological vector spaces (TVS) that generalize normed spaces.
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Mathematical analysis
Mathematical analysis is the branch of mathematics dealing with limits and related theories, such as differentiation, integration, measure, infinite series, and analytic functions.
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Metric space
In mathematics, a metric space is a set for which distances between all members of the set are defined.
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Net (mathematics)
In mathematics, more specifically in general topology and related branches, a net or Moore–Smith sequence is a generalization of the notion of a sequence.
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Normed vector space
In mathematics, a normed vector space is a vector space over the real or complex numbers, on which a norm is defined.
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Nowhere dense set
In mathematics, a nowhere dense set on a topological space is a set whose closure has empty interior.
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P-adic number
In mathematics, the -adic number system for any prime number extends the ordinary arithmetic of the rational numbers in a different way from the extension of the rational number system to the real and complex number systems.
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Pacific Journal of Mathematics
The Pacific Journal of Mathematics (ISSN 0030-8730) is a mathematics research journal supported by a number of American, Asian and Australian universities and research institutes, and currently published on their behalf by Mathematical Sciences Publishers, a non-profit academic publishing organisation.
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Polish space
In the mathematical discipline of general topology, a Polish space is a separable completely metrizable topological space; that is, a space homeomorphic to a complete metric space that has a countable dense subset.
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Prime number
A prime number (or a prime) is a natural number greater than 1 that cannot be formed by multiplying two smaller natural numbers.
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Product topology
In topology and related areas of mathematics, a product space is the cartesian product of a family of topological spaces equipped with a natural topology called the product topology.
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Pseudometric space
In mathematics, a pseudometric space is a generalized metric space in which the distance between two distinct points can be zero.
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Rational number
In mathematics, a rational number is any number that can be expressed as the quotient or fraction of two integers, a numerator and a non-zero denominator.
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Real number
In mathematics, a real number is a value of a continuous quantity that can represent a distance along a line.
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Sequence
In mathematics, a sequence is an enumerated collection of objects in which repetitions are allowed.
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Set (mathematics)
In mathematics, a set is a collection of distinct objects, considered as an object in its own right.
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Square root of 2
The square root of 2, or the (1/2)th power of 2, written in mathematics as or, is the positive algebraic number that, when multiplied by itself, gives the number 2.
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Subtraction
Subtraction is an arithmetic operation that represents the operation of removing objects from a collection.
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Topological group
In mathematics, a topological group is a group G together with a topology on G such that the group's binary operation and the group's inverse function are continuous functions with respect to the topology.
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Topological space
In topology and related branches of mathematics, a topological space may be defined as a set of points, along with a set of neighbourhoods for each point, satisfying a set of axioms relating points and neighbourhoods.
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Topological vector space
In mathematics, a topological vector space (also called a linear topological space) is one of the basic structures investigated in functional analysis.
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Topology
In mathematics, topology (from the Greek τόπος, place, and λόγος, study) is concerned with the properties of space that are preserved under continuous deformations, such as stretching, crumpling and bending, but not tearing or gluing.
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Total order
In mathematics, a linear order, total order, simple order, or (non-strict) ordering is a binary relation on some set X, which is antisymmetric, transitive, and a connex relation.
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Totally bounded space
In topology and related branches of mathematics, a totally bounded space is a space that can be covered by finitely many subsets of every fixed "size" (where the meaning of "size" depends on the given context).
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Uniform continuity
In mathematics, a function f is uniformly continuous if, roughly speaking, it is possible to guarantee that f(x) and f(y) be as close to each other as we please by requiring only that x and y are sufficiently close to each other; unlike ordinary continuity, the maximum distance between f(x) and f(y) cannot depend on x and y themselves.
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Uniform norm
In mathematical analysis, the uniform norm (or sup norm) assigns to real- or complex-valued bounded functions f defined on a set S the non-negative number This norm is also called the supremum norm, the Chebyshev norm, or the infinity norm. The name "uniform norm" derives from the fact that a sequence of functions \ converges to f under the metric derived from the uniform norm if and only if f_n converges to f uniformly.
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Uniform space
In the mathematical field of topology, a uniform space is a set with a uniform structure.
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Union (set theory)
In set theory, the union (denoted by ∪) of a collection of sets is the set of all elements in the collection.
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Uniqueness quantification
In mathematics and logic, the phrase "there is one and only one" is used to indicate that exactly one object with a certain property exists.
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Unit interval
In mathematics, the unit interval is the closed interval, that is, the set of all real numbers that are greater than or equal to 0 and less than or equal to 1.
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Universal property
In various branches of mathematics, a useful construction is often viewed as the “most efficient solution” to a certain problem.
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Up to
In mathematics, the phrase up to appears in discussions about the elements of a set (say S), and the conditions under which subsets of those elements may be considered equivalent.
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Banach completion, Cauchy complete, Cauchy complete space, Cauchy completeness, Cauchy completion, Complete (topology), Complete metric, Complete space, Complete topological vector space, Completeness (in topology), Completeness (topology), Completion (metric space), Completion (topology), Completion of a metric space, Metric space completion.
References
[1] https://en.wikipedia.org/wiki/Complete_metric_space
